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We derive new exact charged $d$-dimensional black hole solutions forquadratic teleparallel equivalent gravity, $f({\cal T})=a_0+a_1{\calT}+a_2{\cal T}^2$, where $\cal T$ is the torsion scalar, in the case ofnon-linear electrodynamics. We give a specific form of electromagnetic functionand find out the form of the unknown functions that characterize the vielbeinsin presence of the electromagnetic field. It is possible to show that the blackholes behave asymptotically as AdS solutions and contain, in addition to themonopole and quadrupole terms, other higher order terms whose source is thenon-linear electrodynamics field. We calculate the electromagnetic Maxwellfield and show that our d-dimensional black hole solutions coincide with theprevious obtained one \cite{2017JHEP...07..136A}. The structure of thesolutions show that there is a central singularity that is much mild incomparison with the respective one in General Relativity. Finally, thethermodynamical properties of the solutions are investigated by calculating theentropy, the Hawking temperature, the heat capacity, and other physicalquantities. The most important result of thermodynamics is that the entropy isnot proportional to the area of the black hole. This inanition points out thatwe must have a constrain on the quadrupole term to get a positive entropyotherwise we get a negative value.

Rotating and non-rotating AdS black holes in $f({\cal T})$ gravity non-linear electrodynamics